Circles $C_1,C_2$ have radii $3$ and $7$ respectively. The circles intersect at distinct $A$ and $B$. A point $P$ outside $C_2$ lies on the line determined by $A$ and $B$ at a distance $5$ from center $C_1$. Point $Q$ is chosen on $C_2$ such that $PQ$ is tangent at $Q$. Find $PQ$.
What I Tried: Here is a picture in Geogebra :-
I have no good idea on how to start this. I cannot, for example, find the lengths of $C_1C$ and $CC_2$ from pure deduction, ($C_1$ and $C_2$ are centres of the circles) and even if Pythagorean Theorem may look like they can work, I cannot understand how, because I don't have many required lengths. I also tried similarity but in fact I could not find any similar triangles here.
Can anyone help me? Thank You.
Best Answer
$$ |PQ|^2=|PA||PB|= (|PC_1|+r_1)(|PC_1|-r_1) $$